A time series approach to the study of the simple subcritical Galton–Watson process with immigration

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Abstract
The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson processXwith immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems onXare proved leading to the construction of Quenouille-type goodness-of-fit tests forX.It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

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